Fundamentals

An example that highlights connection with ab-initio theory is shown in Fig. 1, Ref. [3], where binding energies calculated exactly for neutron drops confined within the Woods-Saxon potential of two different depths, $V_0=-25$ and $-35.5$MeV, are compared with those corresponding to several Skyrme functionals [4]. Calculations of this type, performed at different depths, surface thicknesses, and deformations of the confining potential may allow for a better determination of Skyrme-functional parameters.

Figure 1: Energies of neutron drops confined by the Woods-Saxon potential, see text. From Ref. [3]; picture courtesy of R.B. Wiringa.

In practice, the exact density functionals (2) or (3) are modelled as integrals of energy densities ${\cal H}$, and thus they are called energy-density functionals (EDFs). They can be local functions of local densities (4), quasilocal functions of local higher-order densities (5), nonlocal functions of local densities (6), or nonlocal functions of nonlocal densities (7).

$$E[\rho(\vec{r})] =\!\!\!\! \int\!\!{\rm d}\vec{r}\,{\cal H}(\rho(\vec{r})),$$ (4)

$$E[\rho(\vec{r})] =\!\!\!\! \int\!\!{\rm d}\vec{r}\,{\cal H}(\rho(\vec{r}),\tau(\vec{r}), \Delta\rho(\vec{r}),\ldots),$$ (5)

$$E[\rho(\vec{r})] =\!\!\!\! \int\!\!{\rm d}\vec{r}\!\!\int\!\!{\rm d}\vec{r}\,'\, {\cal H}(\rho(\vec{r}),\rho(\vec{r}\,')),$$ (6)

$$E[\rho(\vec{r},\vec{r}\,')] =\!\!\!\! \int\!\!{\rm d} \vec{r}\!\!\int\!\!{\rm d}\vec{r}\,'\,{\cal H}(\rho(\vec{r},\vec{r}\,')).$$ (7)

Two major classes of approach that are currently used and developed in nuclear structure physics are based on relativistic and nonrelativistic EDFs [5,4,6]. The nonrelativistic EDFs are most often built as:

$${\cal H}(\rho(\vec{r},\vec{r}\,')) = {1\over 2}\sum_{vt} \le... ...{r}\,')\rho^t_v(\vec{r}\,',\vec{r})\right]^0\right]_0\right) ,$$ (8)

where $\hat{V}^{\mbox{\rm {\scriptsize {dir}}}}_{vt}$ and $\hat{V}^{\mbox{\rm {\scriptsize {exc}}}}_{vt}$ denote the EDF-generating (pseudo)potentials in the direct and exchange channels, respectively, and we sum over the spin-rank $v=0$ and 1 (scalar and vector) and isospin-rank $t=0$ and 1 (isoscalar and isovector) spherical-tensor densities [7,8] coupled to the total isoscalar (superscript 0) and scalar (subscript 0) term. For example, finite-range momentum-independent central potentials generate the Gogny [9] or M3Y [10] nonlocal functionals (8) and zero-range momentum-dependent pseudopotentials generate the Skyrme [4] or BCP [11] quasilocal functionals (6).

Expression (8) derives from the Hartree-Fock formula for the average energy of a Slater determinant. However, the EDF-generating pseudopotentials should not be confused with the nucleon-nucleon (NN) bare or effective interaction or Brueckner G-matrix. Indeed, their characteristic features are different - they neither are meant to describe the NN scattering properties, as the bare NN force is, nor are meant to be used in a restricted phase space, as the effective interaction is, nor depend on energy, as the G-matrix does. Moreover, to ensure correct saturation properties, the EDF-generating pseudopotentials must themselves depend on the density. But most importantly, the generated EDFs are modelled so as to describe the exact binding energies and not those in the Hartree-Fock approximation, which otherwise would have required adding higher-order corrections based on the many-body perturbation theory.

Only very recently, it has been demonstrated [12,8] that the nuclear nonlocal EDFs, based on sufficiently short-range EDF-generating pseudopotentials, are equivalent to quasilocal EDFs. In Fig. 2 are compared the proton RMS radii and binding energies of doubly magic nuclei, determined by using the Gogny D1S EDF [13] and second-order Skyrme-like EDF S1Sb [12] derived therefrom by using the Negele-Vautherin (NV) density-matrix expansion (DME) [14]. One can see that already at second order, the DME gives excellent precision of the order of 1%. In Ref. [15], similar conclusions were also reached when comparing the nonlocal and quasilocal relativistic EDFs, see Fig. 3.

Figure 2: Proton RMS radii (left) and binding energies (right) of doubly magic nuclei, determined by using the Gogny D1S EDF [13] (squares) and second-order Skyrme-like EDF [12] derived therefrom (circles). The insets show results in expanded scales and scaled by particle numbers $A$.

Figs. 4 and 5 show convergence of the direct and exchange interaction energies, respectively, when the Taylor and damped Taylor (DT) DMEs are performed up to sixth order [8]. The four panels of Fig. 5 show results obtained in the four spin-isospin channels labeled by $V_{vt}$. Results of the DT DME [8] are compared with those corresponding to the NV [14] and PSA [16] expansions. It is extremely gratifying to see that in each higher order the precision increases by a large factor, which is characteristic to a rapid power-law convergence. The success and convergence of the DME expansions relies on the fact that the finite-range nuclear effective interactions are very short-range as compared to the spatial variations of nuclear densities. The quasilocal (gradient) expansion in nuclei works!

Figure 3: The relative deviations (in percentages) between the experimental and theoretical charge radii (upper panel) and binding energies (lower panel) of 12 spherical nuclei, calculated with the meson-exchange interaction DD-ME2 and the four point-coupling parameter sets. In all cases, the relative deviations are below 1%. Starting from the meson-exchange density-dependent interaction DD-ME2, the equivalent point-coupling parametrization for the effective Lagrangian was derived by incorporating the density dependence of the parameters. The parameters of the point-coupling model were adjusted to reproduce the nuclear matter equation of state obtained with the DD-ME2 interaction and surface thickness and surface energy of semi-infinite nuclear matter. Reprinted figure with permission from Ref. [15]. Copyright 2008 by the American Physical Society.

Figure 4: Precision of the Taylor DME of the direct interaction energies calculated for the nonlocal Gogny D1S EDF [17]. The nine nuclei used for the test are $^4$He, $^{16}$O, $^{40,48}$Ca, $^{56,78}$Ni, $^{100,132}$Sn, and $^{208}$Pb. From Ref. [8].

Figure 5: The RMS deviations between the exact and approximate exchange energies calculated for the nine nuclei listed in the caption of Fig. 4, see text. From Ref. [8].